Fourth Moment Theorems for Markov Diffusion Generators

Inspired by the insightful article arXiv:1210.7587, we revisit the Nualart-Peccati-criterion arXiv:math/0503598 (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diffusion generators. We are not only able to drastically simplify all of its previous proofs, but also to provide new settings of diffusive generators (Laguerre, Jacobi) where such a criterion holds. Convergence towards gamma and beta distributions under moment conditions is also discussed.Comment: 15 page

Four moments theorems on Markov chains

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumptionwemake on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carré du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.First author draf

Optimal Convergence Rates and One-Term Edgeworth Expansions for Multidimensional Functionals of Gaussian Fields

We develop techniques for determining the exact asymptotic speed of convergence in the multidimensional normal approximation of smooth functions of Gaussian fields. As a by-product, our findings yield exact limits and often give rise to one-term generalized Edgeworth expansions increasing the speed of convergence. Our main mathematical tools are Malliavin calculus, Stein's method and the Fourth Moment Theorem. This work can be seen as an extension of the results of arXiv:0803.0458 to the multi-dimensional case, with the notable difference that in our framework covariances are allowed to fluctuate. We apply our findings to exploding functionals of Brownian sheets, vectors of Toeplitz quadratic functionals and the Breuer-Major Theorem

Four moments theorems on Markov chaos

We obtain quantitative Four Moments Theorems establishing convergence of the laws of elements of a Markov chaos to a Pearson distribution, where the only assumption we make on the Pearson distribution is that it admits four moments. While in general one cannot use moments to establish convergence to a heavy-tailed distributions, we provide a context in which only the first four moments suffices. These results are obtained by proving a general carr\'e du champ bound on the distance between laws of random variables in the domain of a Markov diffusion generator and invariant measures of diffusions. For elements of a Markov chaos, this bound can be reduced to just the first four moments.Comment: 24 page

Gómez Camacho: a paradoxical prophet in 16th century Spain

Gómez Camacho, visionario activo en el Reino de Sevilla entre 1540 y 1553, tiene una relación muy estrecha con los evangélicos sevillanos como Egidio y Constantino, que serian perseguidos en años posteriores. Es un ejemplo claro de un profetismo cercano al de su contemporáneo GuilIaume Postel en Italia, pero, sobre todo, su memoria sobrevivió más de den años en forma de una congregación semiclandestina, ignorada por la Inquisición hasta 1623.Gómez Camacho, visionary active in the Kingdom of Seville between 1540 and 1553, had a very close relationship with the Sevilian evangelical preachers as Egidio and Constantino, who were persecuted in later years, and was an outstanding example of a kind of prophetic charisma Iikely to that of his contemporary Guillaume Postel, in Italy. But, more important, his memory survived for more than a century by way of a semiclandestine congregation, ignored by the Inquisition until 1623

Gómez Camacho: un profeta paradójico en el Siglo de Oro

Gómez Camacho, visionario activo en el Reino de Sevilla entre 1540 y 1553, tiene una relación muy estrecha con los evangélicos sevillanos como Egidio y Constantino, que serian perseguidos en años posteriores. Es un ejemplo claro de un profetismo cercano al de su contemporáneo GuilIaume Postel en Italia, pero, sobre todo, su memoria sobrevivió más de den años en forma de una congregación semiclandestina, ignorada por la Inquisición hasta 1623.Gómez Camacho, visionary active in the Kingdom of Seville between 1540 and 1553, had a very close relationship with the Sevilian evangelical preachers as Egidio and Constantino, who were persecuted in later years, and was an outstanding example of a kind of prophetic charisma Iikely to that of his contemporary Guillaume Postel, in Italy. But, more important, his memory survived for more than a century by way of a semiclandestine congregation, ignored by the Inquisition until 1623

Notch and NF-kB: Coach and Players of Regulatory T-Cell Resposnse in Cancer

The Notch signaling pathway plays multiple roles in driving T-cell fate decisions, proliferation, and aberrant growth. NF-kB is a cell-context key player interconnected with Notch signaling either in physiological or in pathological conditions. This review focuses on how themultilayered crosstalk between different Notches and NF-kB subunits may converge on Foxp3 gene regulation and orchestrate CD4+ regulatory T (Treg) cell function, particularly in a tumor microenvironment. Notably, Treg cells may play a pivotal role in the inhibition of antitumor immune responses, possibly promoting tumor growth. A future challenge is represented by further dissection of both Notch and NF-kB pathways and consequences of their intersection in tumor-associated Treg biology. This may shed light on themolecularmechanisms regulating Treg cell expansion andmigration to peripheral lymphoid organs thought to facilitate tumor development and still to be explored. In so doing, new opportunities for combined and/or more selective therapeutic Q25 approaches to improve anticancer immunity may be found

Limit theorems for general functionals of Brownian local times

In this paper, we present the asymptotic theory for integrated functions of increments of Brownian local times in space. Specifically, we determine their first-order limit, along with the asymptotic distribution of the fluctuations. Our key result establishes that a standardized version of our statistic converges stably in law towards a mixed normal distribution. Our contribution builds upon a series of prior works by S. Campese, X. Chen, Y. Hu, W.V. Li, M.B. Markus, D. Nualart and J. Rosen \cite{C17,CLMR10,HN09,HN10,MR08,R11,R11b}, which delved into special cases of the considered problem, such as quadratic, cubic and polynomial cases. We establish the limit theorem for general functions that satisfy mild smoothness and growth conditions. This extends the scope beyond the polynomial cases studied in previous works, providing a more comprehensive understanding of the asymptotic properties of the considered functionals